Complex structure on $\mathbb S^1 \times \mathbb S^3$

Question

Apparently there is a complex structure on $X = \mathbb S^1 \times \mathbb S^3$ but I have no ideas why. Any hints ?

Answer 1

There are many. Take $X=\mathbb C^2 \backslash \{(0,0)\}$, and take $A \in GL_2(\mathbb C)$ such that all eigenvalues of $A$ are strictly less than one.

It can be proved that $S:=X/\langle A \rangle$ is diffeomorphic to $S^1 \times S^3$ (in fact, Kodaira even proved the converse: any complex manifold diffeomorphic to $S^1 \times S^3$ can be constructed in this way [edit: or something very very similar]). It can also be proved that two such complex surfaces $S_1$ and $S_2$ are biholomorphic if and only if $A_1$ and $A_2$ are conjugate.

Google "Hopf surface" for more information.